http://www.ck12.org Chapter 5. Relationships with Triangles
As you might guess, the largest angle will be opposite 18 because it is the longest side. Similarly, the smallest angle
will be opposite the shortest side, 7. Therefore, the angle measure in the middle will be opposite 13.
Theorem 5-9:If one side of a triangle is longer than another side, then the angle opposite the longer side will be
larger than the angle opposite the shorter side.
Converse of Theorem 5-9:If one angle in a triangle is larger than another angle in a triangle, then the side opposite
the larger angle will be longer than the side opposite the smaller angle.
Proof of Theorem 5-9
Given:AC>AB
Prove:m^6 ABC>m^6 C
TABLE5.6:
Statement Reason
1.AC>AB Given- Locate pointPsuch thatAB=AP Ruler Postulate
- 4 ABPis an isosceles triangle Definition of an isosceles triangle
4.m^61 =m^63 Base Angles Theorem
5.m^63 =m^62 +m^6 C Exterior Angle Theorem
6.m^61 =m^62 +m^6 C Substitution PoE
7.m^6 ABC=m^61 +m^62 Angle Addition Postulate
8.m^6 ABC=m^62 +m^62 +m^6 C Substitution PoE
9.m^6 ABC>m^6 C Definition of “greater than” (from step 8)
To prove the converse, we will need to do so indirectly. This will be done in the extension at the end of this chapter.
Example 1:List the sides in order, from shortest to longest.